Mean Deviation
المؤلف:
Kenney, J. F. and Keeping, E. S.
المصدر:
"Mean Absolute Deviation." §6.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand
الجزء والصفحة:
...
20-2-2021
2071
Mean Deviation
The mean deviation (also called the mean absolute deviation) is the mean of the absolute deviations of a set of data about the data's mean. For a sample size 
, the mean deviation is defined by
 |
(1)
|
where 
is the mean of the distribution. The mean deviation of a list of numbers is implemented in the Wolfram Language as MeanDeviation[data].
The mean deviation for a discrete distribution 
defined for 
, 2, ..., 
is given by
 |
(2)
|
Mean deviation is an important descriptive statistic that is not frequently encountered in mathematical statistics. This is essentially because while mean deviation has a natural intuitive definition as the "mean deviation from the mean," the introduction of the absolute value makes analytical calculations using this statistic much more complicated than the standard deviation
 |
(3)
|
As a result, least squares fitting and other standard statistical techniques rely on minimizing the sum of square residuals instead of the sum of absolute residuals.
For example, consider the discrete uniform distribution consisting of 
possible outcomes with 
for 
, 2, ..., 
. The mean is given by
 |
(4)
|
The variance (and therefore its square root, namely the standard deviation) is also straightforward to obtain as
 |
(5)
|
On the other hand, the mean deviation is given by
 |
(6)
|
This can be obtained in closed form, but is much more unwieldy since it requires breaking up the summand into two pieces and treating the cases of 
even and odd separately.
The following table summarizes the mean absolute deviations for some named continuous distributions, where 
is an incomplete beta function, 
is a beta function, 
is a gamma function, 
is the Euler-Mascheroni constant, 
is a Meijer G-function, 
is the exponential integral function, 
is erf, and 
is erfc.
| distribution |
M.D. |
| beta distribution |
 |
| chi-squared distribution |
 |
| exponential distribution |
 |
| gamma distribution |
 |
| Gumbel distribution |
![beta[G_(2,1)^(2,0)(e,gamma^(-1)|1,1; 0)-Ei(sinhgamma-coshgamma)]](https://mathworld.wolfram.com/images/equations/MeanDeviation/Inline23.gif) |
| half-normal distribution |
![2/theta[e^(-1/pi)+erf(1/(sqrt(pi)))]](https://mathworld.wolfram.com/images/equations/MeanDeviation/Inline24.gif) |
| Laplace distribution |
 |
| logistic distribution |
 |
| log normal distribution |
 |
| Maxwell distribution |
![4e^(-4/pi)sqrt(2/pi)sigma[1+e^(4/pi)erf(2/(sqrt(pi)))]](https://mathworld.wolfram.com/images/equations/MeanDeviation/Inline28.gif) |
| normal distribution |
 |
| Pareto distribution |
 |
| Rayleigh distribution |
 |
| Student's t-distribution |
 |
| Student's t-distribution |
 |
| triangular distribution |
 |
| triangular distribution |
![<span style=]() {(2(b+c-2a)^3)/(81(a-b)(a-c)) for a+b<2c; (2(a+c-2b)^3)/(81(a-b)(b-c)) for a+b>2c" src="https://mathworld.wolfram.com/images/equations/MeanDeviation/Inline35.gif" style="height:66px; width:197px" /> |
| uniform distribution |
 |
The following table summarizes the mean absolute deviations for some named discrete distributions, where 
.
| distribution |
M.D. |
| Bernoulli distribution |
 |
| binomial distribution |
 |
| discrete uniform distribution |
![<span style=]() {1/4N for N even; ((N-1)(N+1))/(4N) for N odd" src="https://mathworld.wolfram.com/images/equations/MeanDeviation/Inline40.gif" style="height:58px; width:154px" /> |
| geometric distribution |
 |
| Poisson distribution |
 |
| Zipf distribution |
![(2[zeta(rho+1)zeta(rho,|_mu_|+1)-zeta(rho)zeta(rho+1,|_mu_|+1)])/(zeta^2(rho+1))](https://mathworld.wolfram.com/images/equations/MeanDeviation/Inline43.gif) |
REFERENCES:
Havil, J. "Ways of Means." §13.1 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 119-121, 2003.
Kenney, J. F. and Keeping, E. S. "Mean Absolute Deviation." §6.4 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 76-77 1962.
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